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Properties

Label 1-407-407.147-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.999 + 0.0237i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.309 + 0.951i)6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + 10-s + 12-s + (−0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + (0.809 − 0.587i)18-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.309 + 0.951i)6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + 10-s + 12-s + (−0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + (0.809 − 0.587i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.999 + 0.0237i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.999 + 0.0237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002110628834 - 0.1775499624i\)
\(L(\frac12)\) \(\approx\) \(0.002110628834 - 0.1775499624i\)
\(L(1)\) \(\approx\) \(0.4278048331 - 0.1793822288i\)
\(L(1)\) \(\approx\) \(0.4278048331 - 0.1793822288i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (0.809 + 0.587i)T \)
23 \( 1 - T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.57781911269115034483789235907, −23.81710989303063981820948963387, −23.263227718506290116731997093012, −22.36311706224904151805438484559, −21.57883199806401421271464002701, −20.228480147313882252506098038479, −19.6117915068406492526666022218, −18.31880767075794552470078429733, −17.51548558181471762863090880102, −16.54356012571637284865309056783, −16.20381207090719112123123092242, −15.60981392961650328524871772519, −14.249895618955374472709032032498, −13.35588421882193319670522605313, −12.29458683884317391345167413772, −11.326819046798413300843231894919, −9.949563046354508927711126704137, −9.509337960690561678646211033595, −8.523977533435276779064724882070, −7.13417727695561438403651455828, −6.53462783747107209909489260949, −5.224975957159706851129701125879, −4.64824562909678188474815002750, −3.63274682446272490617816462567, −1.11849806218646706333948262920, 0.15075163949677759696262908579, 1.89433836287288513605915984564, 2.89275011451132303084129506159, 3.93252313631710271112484536983, 5.45810205262632445404732126006, 6.361213570781790992770373783296, 7.54795850361859046252225312681, 8.35110656011552928898243702987, 10.001487301192600315614337277391, 10.311997991804356551427637401439, 11.51937487352079177822805047639, 12.08741091918708490516070892168, 12.931167021953588143494991970386, 13.79197475529953359816471132205, 15.11367162633295298822713422022, 16.159684432812813435111602771801, 17.24683668355308368029455067244, 18.09032077058186375126306903178, 18.63689098441793118966007570611, 19.41609355484291675042181660561, 20.10571709539514796301354675164, 21.64862026468771151398635217796, 22.215319663753950955477844798524, 22.72452346242757261896617509792, 23.53392653314765028405161224258

Graph of the $Z$-function along the critical line